2018 Theses Doctoral

Quantum K-theory and the Baxter Operator

Pushkar, Petr

In this work, the connection between quantum K-theory and quantum integrable systems is studied. Using quasimap spaces the quantum equivariant K-theory of Naka- jima quiver varieties is defined. For every tautological bundle in the K-theory there exists its one-parametric deformation, referred to as quantum tautological bundle. For specific cases of cotangent bundles to Grassmannians and flag varieties it is proved that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous XXZ spin chain. It is also proved that each such operator corresponds to the universal elements of quantum group U􏰁(sln). In particular, the Baxter operator for the XXZ spin chain is identified with the operator of quantum multiplication by the exterior algebra of the tautological bundle. An explicit universal combinatorial formula for this operator is found in the case of U􏰁(sl2). The relation between quantum line bundles and quantum dynamical Weyl group is shown. This thesis is based on works [37] and [24].